Memoire d’habilitation a diriger des recherchesUniversite Lille 1

Laboratoire Paul Painleve

Florent BALACHEFF

Contributions a la geometrie systolique locale et globale

My research is devoted to systolic geometry and related topics. In this report, I will present themain results corresponding to the following selected papers:

[1] Contact geometry and isosystolic inequalities.With J.-C. Alvarez Paiva.Preprint (2011).

[2] Distribution of the systolic volume of homology classes.With I. Babenko.Preprint (2010).

[3] Bers’ constants for punctured spheres and hyperelliptic surfaces.With H. Parlier.Journal of Topology and Analysis, Vol. 4 (2012), No 3, 271-296.

[4] Short loop decompositions of surfaces and the geometry of Jacobians.With H. Parlier and S. SabourauGeometric And Functional Analysis, Vol. 32 (2012), No 1, 37-73.

[5] A local optimal diastolic inequality on the two-sphere.Journal of Topology and Analysis, Vol. 2 (2010), No 1, 109-121.

[6] Diastolic inequalities and isoperimetric inequalities on surfaces.With S. Sabourau.Annales de l’Ecole Normale Superieure, Vol. 43 (2010), No 4, 579-605.

[7] A Zoll counterexample to a geodesic length conjecture.With C. Croke and M.G. Katz.Geometric and Functional Analysis, Vol. 19 (2009), No 1, 1-10.

[8] Sur la systole de la sphere au voisinage de la metrique standard.Geometriae Dedicata, Vol. 121 (2006), No 1, 61-71.

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Introduction and statement of the results

We start this report with a short overview of the main results we will present in the sequel.First recall the context. Let M be a closed manifold of dimension m. For a given Riemannianmetric g on M , we denote by sys(M, g) the systole defined as the least length of a closed geodesicand by vol(M, g) the Riemannian volume. When the manifold is non-simply connected, we definethe homotopical systole denoted by sysπ(M, g) as the least length of a non-contractible closedgeodesic. Lastly, when the first integral homology group of the manifold is non-trivial, we definethe homological systole denoted by sys1(M, g) as the least length of a homologically non-trivialclosed geodesic. There exist other notions of systole such as the Z2-homological systole, the stablesystole or the higher dimensional systoles for instance, but they will not be considered in the sequel.It is important to underline that the term systole usually stands for the homotopical systole, butin order to unify our terminology, we fix this vocabulary which was suggested by M. Berger in[Berg00, p.107].

The main topic in systolic geometry is to etablish systolic inequalities which are inequalities ofthe form

vol(M, g) ≥ C · sys∗(M, g)m,where C is some positive constant and which hold for a large subset of metrics g on M . Heresys∗ denotes either the systole, the homotopical systole or the homological systole. If we authorizemore general metrics such as Finsler metrics, then the first systolic inequality was discovered byH. Minkowski [Mink96], who proved that the Hausdorff volume of any flat reversible Finsler torus(Tm, F ) of dimension m satisfies the optimal inequality

vol(Tm, F ) ≥ bm2m· sys1(Tm, F )m.

Here bm denotes the Euclidian volume of the unit ball in dimension m. This statement is areformulation of the celebrated theorem on which is based the geometry of numbers theory, andobviously still holds for the homotopical systole or even the systole. Since, other optimal systolicinequalities have been discovered, see [CroKatz02] for an overview.

Variations around systolic inequalities.

In its celebrated paper [Gro83], M. Gromov proved the following central result in systolic ge-ometry: every essential closed manifold satisfies a systolic inequality for the set of all Riemannianmetrics. In the same paper, he also precised the result for closed orientable surfaces of genus γ: aclosed Riemannian surface of area equal to its genus γ admits a homologically non-trivial closedgeodesic of length at most log γ (up to some universal constant). This bound is known to be optimalsince the work [BusSar94] of P. Buser and P. Sarnak.

In the article [4] written in collaboration with H. Parlier and S. Sabourau, we extend Gromov’sasymptotic log γ bound on the homological systole of genus γ surfaces: for any λ ∈ (0, 1) there existsa constant Cλ such that every closed Riemannian surface of genus γ whose area is normalized at γhas at least [λγ] homologically independent loops of length at most Cλ log γ. We construct hyperbolicsurfaces showing that our general result is sharp for the number of such loops. We also extend theupper bound obtained by P. Buser and P. Sarnak in [BusSar94] on the minimal norm of nonzeroperiod lattice vectors of Riemann surfaces in their geometric approach of the Schottky problemto almost γ homologically independent vectors. Finally we derive a lower bound on the systolicarea of finitely presentable groups with no free factor isomorphic to Z in terms of their first Bettinumber which corresponds to a generalization of Gromov’s asymptotic log γ bound.

In the article [6] written in collaboration with S. Sabourau, we present another type of general-ization of Gromov’s result. Loosely speaking, we prove the following result: any closed Riemanniansurface of area equal to its genus γ can be swept out by a family of multi-loops of total length at

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most γ (up to some universal constant). This diastolic inequality, which relies on an upper boundon Cheeger’s constant, yields an effective process to find short closed geodesics on the two-sphere,for instance. The diastolic term comes from the fact that a family of multi-loops sweeping out asurface produces a closed geodesic via a minimax process.

Finally in the article [3] written in collaboration with H. Parlier we consider another type ofminimax quantity which bounds from below the area of surfaces. Our central result is as follows:any hyperbolic two-sphere with n cusps admits a decomposition by three-holed spheres all of whoseboundary loops are of length at most

√n (up to some universal constant). We produce examples

showing that this bound is optimal. These results address to a question asked by P. Buser in[Bus92].

Local versions of systolic inequalities.

Among subsets of metrics for which systolic inequalities are relevant, we find neighbourhoods of afixed metric (for some reasonnable topology). In [Cro88] C. Croke proved the existence of a systolicinequality for the set of all Riemannian metrics on the two-sphere and conjectured (following E.Calabi) that the corresponding optimal constant C should be 1/(2

√3). This optimal constant

corresponds to the flat metric gc with three conical singularities obtained by gluing two copies ofan equilateral triangle along their boundary. In [5], we proved the following local version of thisconjecture: any Riemannian metric g sufficiently C1-closed from the Calabi-Croke metric gc outsideits singularities satisfies the optimal inequality

vol(S2, g) ≥ 12√

3· sys(S2, g)2,

the equality case being reached only by metrics homothetic to gc. It is important to remark that suchRiemannian metrics g necessarily also have three conical singularities of angle 2π/3. The majoringredient in our proof is a degree 3 ramified cover of the two-sphere by the two-torus, relying thesystolic properties of both surfaces.

In the same spirit, we can look for systolic inequalities available for a subset of metrics madeof some one-parameter deformations of a fixed metric. The first result in this direction has beenproved in [8] and states that the round metric g0 on the two-sphere satisfies an optimal systolicinequality for natural one-parameter deformations: for any non-trivial smooth function Ψ : S2 → Rsuch that

∫S2 Ψdvg0 = 0 there exists a positive ε such that for 0 < |t| < ε

area(S2, (1 + tΨ)2 · g0)sys(S2, (1 + tΨ)2 · g0)2

>area(S2, g0)sys(S2, g0)2

=1π.

In the article [1] written in collaboration with Juan-Carlos Alvarez Paiva, we reformulate systolicgeometry in a contact-geometric context. By using the canonical perturbation theory to exploit thelarge symmetry group that systolic geometry inherits, we prove among others things the followingoptimal systolic inequality: any smooth volume-preserving deformation gt of the canonical metricg0 on the real projective space RPm which does not coincide to all orders to trivial defomations ofthe form φ∗t g0 for some isotopy φt satisfies

vol(RPm, gt)sys(RPm, gt)m

>vol(RPm, g0)

sys(RPm, g0)m

for small non-zero values of t. To appreciate the result, recall that for m > 2 the systolic optimalityof the standard metric g0 on RPm is an open question since the work of P. Pu in 1952. In thispaper, we also characterize contact structures which are critical for the systolic volume: they areexactly the regular contact structures, that is contact forms for which the Reeb flow is periodic (theanalog of Zoll metrics in Riemannian geometry). As a consequence, for any closed manifold, the

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smooth Finsler metrics which are critical for the systolic volume associated to the Holmes-Thompsonvolume notion are exactly Zoll Finsler metrics.

The comparaison of the systole with other Riemannian invariants such as the diameter is alsoa natural and deep question. For a closed non-simply connected Riemannian manifold (M, g), itis straighforward to see that sys(M, g) ≤ 2 · diam(M, g). But for simply connected manifolds,the question is much more difficult. Until now, the only simply connected manifold for which auniversal inequality exists between the systole and the diameter is the two-sphere. More precisely,A. Nabutovsky & R. Rotman [NabRot02] and S. Sabourau [Sab04], improving previous boundsdue to C. Croke [Cro88] and M. Maeda [Mae94], proved that for any Riemannian metric g on thetwo-sphere sys(S2, g) ≤ 4 · diam(S2, g). It was a long-standing conjecture that the best constantin this inequality should be 2, the equality case being reached by the round metric. In the article[7] written in collaboration with C. Croke and M. Katz, we prove that this conjecture is false, evenin its local form: there exists smooth deformations gt of the round metric by Zoll Riemannianmetrics such that sys(S2, gt) > 2 · diam(S2, gt) for small non-zero values of t.

Systolic geometry of homology classes.

Given a pair (G, a) where G is a finitely presentable group and a an integer homology class of G,the systolic volume is defined as the least volume of any geometric realisation of the homology classby a pseudomanifold endowed with a polyhedral metric for which the length of curves representingelements of G are at least 1. We know by [Gro83] that an essential manifold M always satisfiesa systolic inequality for the homotopical systole over the set of all Riemannian metrics. The bestconstant S(M) involved in such an inequality is called the systolic constant and coincides withthe systolic volume of the pair (π1(M), f∗[M ]) where f : M → K(π1(M), 1) is the classifyingmap of M into its corresponding Eilenberg-MacLane space. This result shared by I. Babenko[Bab06, Bab08] and M. Brunnbauer [Bru08] determines precisely the topological nature of thesystolic constant. In the paper [2] written in collaboration with I. Babenko we study numerousaspects of the systolic volume showing that it is a complex invariant of homology classes of finitelypresentable groups. Among other results, we show that for any dimension m ≥ 3 there existsan infinite sequence of finitely presentable pairwise distinct groups Gi for which at least oneirreducible class ai ∈ Hm(Gi,Z) satisfies S(Gi, ai) ≤ 1. So we can not hope finiteness resultswithout additional restrictions. An example of efficient restriction is illustrated by the followingresult: the systolic volume of integer multiples ka of a fixed homolgy class a is a sublinear functionin k. This result is strongly connected with the asymptotic behaviour of the systolic constantunder the operation of connected sum. We also prove that the systolic volume of a homology classis bounded from below by its torsion, thus addressing to a question of M. Gromov in [Gro96].

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1. Geometry of surfaces through the length of closed curves

The first part of this report deals with surfaces. In the same way a taylor determines thegeometry of a body by carefully taking some relevant measures, we study the geometry of surfacesthrough the lengths of some special closed curves. These curves are respectively the diastole, whichcorresponds to a curve obtained by a minimax process on the one-cycle space, the curves involvedin a short decomposition of the surface by elementary topological subsets—the pants—, and shorthomologically independent curves. We look for bounds on the length of these curves by the areaof the surface which is equivalent to estimate their length while the area is normalized. We willpresent the main results of the article [3], [4], and [6] written in collaboration with H. Parlierand S. Sabourau. In the sequel, all surfaces are supposed to be orientable in order to facilitatethe presentation and we refer to [3], [4], and [6] for considerations in the non-orientable case. AllRiemannian metrics are supposed to be smooth. Otherwise stated all surfaces are supposed to beclosed and connected.

1.1. Diastole. Using the isomorphism between the second homotopy group of a two-sphere andthe fundamental group of its loop space relatively to the subspace of constant loops, G. D. Birkhoffproved the existence of a nontrivial closed geodesic on every Riemannian two-sphere using a mini-max argument, see [Bir27]. The length of the shortest closed geodesic obtained by such a minimaxprocess can not be uniformly bounded by the square root of the area as follows from the exampleof two-spheres with constant area and three spikes arbitrarily long (see [Sab04, Remark 4.10] forfurther detail). This made remarkable the result of C. Croke [Cro88] that the systole—the shortestlength of a closed geodesic—of any Riemannian two-sphere is uniformly bounded by the squareroot of its area. More precisely, any Riemannian metric g on the two-sphere satisfies the systolicinequality

(1.1) sys(S2, g) ≤ 31√

area(S2, g).

The constant in this inequality is not optimal and has been improved several times, see [NabRot02],[Sab04] and [Rot06].

On non-simply connected closed Riemmanian surfaces, no minimax principle is required to showthe existence of a closed geodesic. The minimum of the length functional over all non-contractibleloops is always positive and realized by the length of a non-contractible closed geodesic. We call thisminimum the homotopical systole and denote it by sysπ. In particular the systole is well definedon such surfaces and satisfies sys ≤ sysπ. This is a central result in systolic geometry that thehomotopical systole can be uniformly bounded by the square root of the area. More precisely,every closed Riemannian surface (M, g) of genus γ ≥ 1 satisfies the following systolic inequality

(1.2) sysπ(M, g) ≤ C log(γ + 1)√γ

√area(M, g)

where C is a universal constant, cf. [Gro83], [Bal04] and [KatzSab05] for three different proofs. Inthis theorem, the homotopical systole can be replaced by the homological systole— the shortestlength of a homologically non-trivial loop—, see [Gro96]. The dependence on the genus in inequal-ity (1.2) is optimal for both homotopical and homological systoles, see [BusSar94].

The existence of a closed geodesic on a Riemannian closed surface (M, g), possibly a two-sphere, can also be proved through a minimax argument on a different space, namely the one-cyclespace Z1(M ; Z). Loosely speaking, this space arising from geometric measure theory is made ofmultiple curves (unions of oriented loops) endowed with some special topology, cf. [6] for a precisedefinition. The use of the one-cycle space, rather than the loop space, introduces some flexibility

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and permits to define a minimax process on any closed Riemannian surface (M, g) using F. Alm-gren’s isomorphism [Almg60] between the relative fundamental group π1(Z1(M ; Z), 0) and thesecond homology group H2(M ; Z) ' Z. More precisely, we define the diastole over the one-cyclespace as

(1.3) dias(M, g) := inf(zt)

sup0≤t≤1

mass(zt)

where (zt) runs over the families of one-cycles inducing a generator of π1(Z1(M ; Z), 0) andmass(zt) represents the mass (or length) of zt. From a result of J. Pitts [Pit74, p. 468], [Pit81,Theorem 4.10] (see also [CalCao92]), this minimax principle gives rise to a union of closed geodesics(counted with multiplicity) of total length dias(M, g). Hence,

sys(M, g) ≤ dias(M, g).

This principle has been used in [CalCao92], [NabRot02], [Sab04], [Rot05] and [Rot06] in the studyof closed geodesics on Riemannian two-spheres.

In the article [6] we show in collaboration with S. Sabourau that the length of the shortest closedgeodesic obtained by such a minimax process on the one-cycle space is uniformly bounded by thesquare root of the area. More precisely, we obtain the following diastolic inequality.

Theorem 1.1. There exists a positive constant C ≤ 108 such that every closed Riemannian sur-face (M, g) of genus γ ≥ 0 satisfies

(1.4) dias(M, g) ≤ C√γ + 1

√area(M, g).

Since the minimax principle (1.3) gives rise to a union of closed geodesics of length dias(M, g),Theorem 1.1 yields a construction of short closed geodesics on surfaces through Morse theory overthe one-cycle space.

The dependence on the genus in inequality (1.4) is optimal: the closed hyperbolic surfacesof arbitrarily large genus with Cheeger constant bounded away from zero constructed in [Bro86]provide examples of surfaces with area ' γ and dias & γ, see [6, Remark 7.3]. This dependence ofthe inequality (1.4) on the genus should be compared with the one in (1.2).

When the metric is bumpy, the diastole corresponds by definition to the length of a one-cycle ofindex one. Recall that the index of a one-cycle z of mass κ is defined as

indZ(z) = mini ∈ N | πi(Zκ1 (M,Z) ∪ γ,Zκ1 (M,Z)) is not trivial

where Zκ1 (M,Z) = z ∈ Z1(M,Z) | mass(z) < κ. It is important to remark that the relationbetween the filling radius (see [Gro83]) of a bumpy Riemannian two-sphere and the length of itsshortest one-cycle of index one established in [Sab04] cannot be extended to the diastole. Indeed,from [Sab04, Theorem 1.6], there exists a sequence gn of Riemannian metrics on the two-spheresuch that

limn→+∞

FillRad(S2, gn)dias(S2, gn)

= 0.

This result illustrates the difference of nature between the length of the shortest closed geodesicor of the shortest one-cycle of index one, which can be bounded by the filling radius on the two-sphere, and the diastole. It also shows that the proof of Theorem 1.1 requires different techniques.The proof relies on an inequality between the area and the Cheeger constant, and a cut-and-pasteargument on one-cycles.

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1.2. Pants decompositions. Start with a hyperbolic surface S of genus γ with n cusps and recallthat its area is equal to 2π(2γ − 2 + n). A pants decomposition is a maximal collection of disjointsimple closed geodesics. Such a collection is necessarily made of exactly 3γ − 3 + n loops whosecomplementary region is a disjoint union of 2γ − 2 + n surfaces of topological type (0, 3) (so-calledpants). For a given pants decomposition P of S, we define its length as

length(P) = maxα∈P

length(α).

L. Bers [Bers74, Bers85] showed that there always exists a pants decomposition whose lengthis bounded from above by some universal constant which only depends on the topology of thesurface. This result was quantified in the closed case by P. Buser [Bus81, Bus92], and P. Buser andM. Seppala [BusSep92] who showed that the optimal constant —called Bers’ constant— behavesat least like ∼ √γ and at most like ∼ γ. In the punctured case P. Buser proved [Bus92] that Bers’constant grows at most linearly in the number of cusps. In any case the correct behavior remainsunknown, but P. Buser conjectured the following.

Conjecture 1.1. Bers’ constants for surfaces of genus γ with n cusps behave roughly like√γ + n.

In the sequel we will denote Bers’ constant in signature (γ, n) by Bγ,n. Before explaining ourresults in the hyperbolic case, we explain how to generalize pants decomposition to the Riemanniancontext. Given a Riemannian surface (M, g) of genus γ with nmarked points, a pants decompositionis a collection of 3γ − 3 + n disjoint simple loops which cut the surface into 2γ − 2 + n pairs ofpants. Here a pair of pants is either a three holed sphere, a cylinder with one marked point or adisk with two marked points. The length of such a pants decomposition is defined as the maximallength of its loops. As a byproduct of the proof of Theorem 1.1 we derive the following result, see[4, Proposition 6.3].

Corollary 1.1. Let (M, g) be a closed Riemannian surface of genus γ with n marked points. Then(M, g) admits a pants decomposition with respect to the marked points of length at most

C√γ + 1

√area(M, g)

for an explicit universal constant C.

This corollary implies previously known linear upperbounds by P. Buser and M. Seppala for genusgrowth. It also implies the square root upperbound conjectured by P. Buser for hyperbolic punc-tured spheres. Indeed given a hyperbolic sphere S with n punctured points, cut a small punctureddisk of area ε around each punctured point and define a new surface S by gluing along each bound-ary component a round hemisphere with a marked point at its center. As the Riemannian surfaceS with n marked points (the centers of the round hemispheres) admits a pants decomposition oflength at most

C

√area(S),

this implies that

B0,n ≤ C√

2π(n− 2)

by letting ε→ 0 as area(S) = 2π(n− 2).

But the constant involved in this last inequality is very bad (more than 107), the techniques of[6] being not adapted to the study of pants decomposition. In collaboration with H. Parlier, weprove in [3] Buser’s conjecture in the case of punctured spheres by showing that it fundamentallyrelies on a classical result: the so-called Besicovitch lemma. Then we prove Buser’s conjecture inthe closed case for hyperelliptic surfaces. More precisely, our results are the following.

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Theorem 1.2. 1) Any hyperbolic sphere with n ≥ 4 cusps admits a pants decomposition of lengthat most 30

√2π(n− 2).

2) For any n > 6, there exists a hyperbolic sphere with n punctured points such that any pantsdecomposition has length at least

8 arcsinh 1 (

√n− 4

2− 1).

3) Any hyperelliptic hyperbolic surface of genus γ ≥ 2 admits a pants decomposition of length atmost 51

√4π(γ − 1).

4) For any γ ≥ 2, there exists a hyperelliptic hyperbolic surface of genus γ such that any pantsdecomposition has length at least

4arcsinh1(

√γ − 3

2− 1).

In particular, we derive from this theorem the optimal dependance of Bers’ constant B0,n in thenumber of punctured points:

4√n− 2 . B0,n ≤ 76

√n− 2.

The proof of point 1) in Theorem 1.2 relies both on an induction argument and Besicovitch lemma,and should be compared to the proof of Theorem 1.1. The proof of point 2) is the explicit construc-tion of the asymptotically optimal two-spheres. This construction is inspired by the example ofP. Buser called the ”hairy torus”—a hairy torus with hair tips pairwise glued together whose genusis γ and whose length of pants decompositions asymptotically grows at least like ∼ √γ—. Point3) is obtained as follows. Our upper estimate for length of short pants decomposition of puncturedspheres admits a version for two-spheres with conical points of angle π. Quotient of hyperellipticsurfaces by their hyperelliptic involution being such two-spheres, we pull back short pants decom-position on the quotient sphere to pants decomposition of controlled length on the initial surface.Lastly point 4) is proved by adapting the examples of point 2) to quotient of hyperelliptic surfacesand using the strategy of point 3).

Given a pants decomposition on a closed marked Riemannian surface or a punctured hyperbolicsurface, we can also ask for bounds on the total length of the pants decomposition defined as thesum of the length of the curves involved in the pants decomposition. Our main theorem in thisdirection states that for any fixed genus γ, one can control the growth rate of total length of somepants decomposition of a surface of area ∼ γ + n by a factor which grows like n log n, where n isthe number of marked or punctured points. This result appeared in [4] written in collaborationwith H. Parlier and S. Sabourau. More precisely, we have the following statement.

Theorem 1.3. Fix n ≥ 1 and γ ≥ 0. Let (M, g) be a hyperbolic surface of genus γ with ncusps or a closed Riemannian surface of genus γ with n marked points whose area is normalized to2π(2γ + n− 2).

Then (M, g) admits a pants decomposition whose total length is bounded from above by

Cγ n log(n+ 1),

where Cγ is an explicit genus dependent constant.

This estimate is sharp except possibly for the log(n + 1) term. Indeed, the total lengths of thepants decompositions of hyperbolic surfaces of genus γ with n cusps and no closed geodesics oflength less than 1

100 are at least C ′γ n for some positive constant C ′γ depending only on the genus.As a corollary to the above we show that a hyperelliptic surface of genus γ admits a pants

decomposition of total length at most ∼ γ log γ. This is in strong contrast with the general case

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according to a result of L. Guth, H. Parlier and R. Young [GutParY12]: “random” hyperbolicsurfaces have all their pants decompositions of total length at least ∼ γ7/6−ε for any ε > 0.

Theorem 1.3 is proved by considering naturally embedded graphs which capture part of thetopology and geometry of the surface, and studying these graphs carefully. This method alsopermits us to generalize estimate (1.2) to almost γ homologically independent curves. This is thepurpose of the next section.

1.3. Homologically independent loops. The homological systole of a closed Riemannian surfaceof genus γ with normalized area 4π(γ − 1) is at most ∼ log γ. This result is equivalent to theversion of inequality (1.2) for the homogical systole due to M. Gromov, see [Gro96, 2.C]. Thedependence in the genus is optimal: there exist families of hyperbolic surfaces, one in each genus,whose homological systoles grow like ∼ log γ. The first of these were constructed by P. Buser andP. Sarnak in their seminal article [BusSar94], and there have been other constructions since by R.Brooks [Bro99] and M. Katz, M. Schaps and U. Vishne [KatzSchVish07]. By showing that theshortest homologically nontrivial loop on a hyperbolic surface lies in a “thick” embedded cylinder,P. Buser and P. Sarnak also derived new bounds on the minimal norm of nonzero period latticevectors of Riemann surfaces. This result paved the way for a geometric approach of the Schottkyproblem which consists in characterizing Jacobians (or period lattices of Riemann surfaces) amongabelian varieties.

In [BusSep02, BusSep03], P. Buser and M. Seppala studied bounds on the lengths of curves ina homology basis for closed hyperbolic surfaces. Note however that without a lower bound onthe homological systole, the γ + 1 shortest homologically independent loop cannot be bounded byany function of the genus. Indeed, consider a hyperbolic surface with γ very short homologicallyindependent (and thus disjoint) loops. Every loop homologically independent from these shortcurves must cross one of them, and via the collar lemma, can be made arbitrarily large by pinchingour initial γ curves.

On the other hand, without assuming any lower bound on the homological systole, M. Gro-mov [Gro83, 1.2.D’] proved that on every closed Riemannian surface of genus γ with area nor-malized to 4π(γ − 1), the length of the γ shortest homologically independent loops is at most∼ √γ. Furthermore, Buser’s so-called hairy torus example [Bus81, Bus92] shows that this boundis optimal, even for hyperbolic surfaces.

In the article [4], we obtain with H. Parlier and S. Sabourau new bounds on the lengths of shorthomology basis for closed Riemannian surfaces with homological systole bounded from below.

Theorem 1.4. Let (M, g) be a closed Riemannian surface of genus γ with homological systole atleast ` and area equal to 4π(γ − 1). Then there exist 2γ loops α1, . . . , α2γ on M which induce abasis of H1(M ; Z) such that

(1.5) length(αk) ≤ C0log(2γ − k + 2)

2γ − k + 1γ,

where C0 = 216

min1,` .In particular:

(1) the lengths of the αi are bounded by C0 γ;(2) the median length of the αi is bounded by C0 log(γ + 1).

The linear upper bound in the genus of item (1) already appearedin [BusSep03] for hyperbolic surfaces, where the authors obtained a similar bound for the length ofso-called canonical homology basis. They also constructed a genus γ hyperbolic surface all of whosehomology bases have a loop of length at least C γ for some positive constant C. This shows that

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the linear upper bound in (1) is roughly optimal. However, the general bound (1.5) on the lengthof the loops of a short homology basis, and in particular the item (2), cannot be derived from thearguments of [BusSep03] even in the hyperbolic case. The bound obtained in (2) is also roughlyoptimal. Indeed, the Buser-Sarnak surfaces [BusSar94] have their homological systole greater orequal to 4

3 log γ minus a constant.

A natural question is to find out for how many homologically independent curves does Gromov’slog γ bound hold. In [4], we show using Theorem 1.4 that on every closed Riemannian surface ofgenus γ with normalized area there exist almost γ homologically independent loops of lengths atmost ∼ log γ. More precisely, we prove the following.

Theorem 1.5. Let η : N→ N be a function such that

λ := supγ

η(γ)γ

< 1.

Then there exists a constant Cλ such that for every closed Riemannian surface (M, g) of genus γthere are at least η(γ) homologically independent loops α1, . . . , αη(γ) which satisfy

length(αi) ≤ Cλlog(γ + 1)√γ

√area(M, g)

for every i ∈ 1, . . . , η(γ).

Typically, this result applies to η(γ) = [λγ] where λ ∈ (0, 1).

Thus, the previous theorem generalizes Gromov’s log bound on the homological systole to thelengths of almost γ homologically independent loops. Note that its proof differs from other systolicinequality proofs. Specifically, it directly yields a log γ bound on the homological systole withoutconsidering the homotopical systole (that is, the shortest length of a homotopically nontrivial loop).Initially, M. Gromov obtained his bound from the bound (1.2) on the homotopical systole usingsurgery, cf. [Gro96, 2.C]. However the original proof of the log γ bound on the homotopical systole,cf. [Gro83, 6.4.D’] and [Gro96], as well as the alternative proofs available, cf. [Bal04, KatzSab05],do not directly apply to the homological systole.

One can ask how far from being optimal our result on the number of short (homologicallyindependent) loops is. Of course, in light of the Buser-Sarnak examples, one can not hope to do(roughly) better than a logarithmic bound on their lengths, but the question on the number ofsuch curves remains. Now, because of Buser’s hairy torus example, we know that the γ shortesthomologically independent loops of a hyperbolic surface of genus γ can grow like ∼ √γ and thatthe result of Theorem 1.5 cannot be extended to η(γ) = γ. Still, one can ask for γ−1 homologicallyindependent loops of lengths at most ∼ log γ, or more generally for any number of homologicallyindependent loops of lengths at most ∼ log γ which grows asymptotically like γ. Note that thesurface constructed from Buser’s hairy torus does not provide a counterexample in any of thesecases.

Our next theorem shows this is impossible, which proves that the result of Theorem 1.5 on thenumber of homologically independent loops whose lengths satisfy a log γ bound is optimal. Beforestating this theorem, it is convenient to introduce the following definition.

Definition 1.1. Given k ∈ N∗, the k-th homological systole of a closed Riemannian mani-fold (M, g), denoted by sysk(M, g), is defined as the smallest real L ≥ 0 such that there exist khomologically independent loops on M of length at most L.

11

With this definition, under the assumption of Theorem 1.5 every closed Riemannian surface ofgenus γ with area 4π(γ − 1) satisfies

sysη(γ)(M, g) ≤ Cλ log(γ + 1)

for some constant Cλ depending only on λ. Furthermore, still under the assumption of Theorem 1.5,Gromov’s sharp estimate, cf. [Gro83, 1.2.D’], with this notation becomes

sysγ(M, g) ≤ C√γwhere C is a universal constant.

We can now state the second main result of [4].

Theorem 1.6. Let η : N→ N be a function such that

limγ→∞

η(γ)γ

= 1.

Then there exists a sequence of genus γk hyperbolic surfaces Sγkwith γk tending to infinity such

that

limk→∞

sysη(γk)(Sγk)

log γk=∞.

We now present an application of theorem 1.5 to the geometry of Jacobians of Riemann surfaces,extending the work [BusSar94] of P. Buser and P. Sarnak.

Consider a closed Riemann surface S of genus γ. We define the L2-norm |.|L2 , simply noted |.|,on H1(S; R) ' R2γ by setting

(1.6) |Ω|2 = infω∈Ω

∫Sω ∧ ∗ω

where ∗ is the Hodge star operator and the infimum is taken over all the closed one-forms ω on Srepresenting the cohomology class Ω. The infimum in (1.6) is attained by the unique closed harmonicone-form in the cohomology class Ω. The space H1(S; Z) of the closed one-forms on S with integralperiods (that is, whose integrals over the cycles of M are integers) is a lattice of H1(S; R). TheJacobian J of S is a principally polarized abelian variety isometric to the flat torus

T2γ ' H1(S; R)/H1(S; Z)

endowed with the metric induced by |.|.

In their geometric approach of the Schottky problem, P. Buser and P. Sarnak [BusSar94] alsoproved that the homological systole of the Jacobian of a Riemann surface S of genus γ is at most∼√

log γ and this bound is optimal. In other words, there is a nonzero lattice vector in H1(S; Z)whose L2-norm satisfies a

√log γ upper bound. We extend their result by showing that there exist

almost γ linearly independent lattice vectors whose norms satisfy a similar upper bound. Moreprecisely, we have the following.

Corollary 1.2. Let η : N→ N be a function such that

λ := supγ

η(γ)γ

< 1.

Then there exists a constant Cλ such that for every closed Riemann surface S of genus γ there areat least η(γ) linearly independent lattice vectors Ω1, . . . ,Ωη(γ) ∈ H1(S; Z) which satisfy

(1.7) |Ωi|2L2 ≤ Cλ log(γ + 1)

for every i ∈ 1, . . . , η(γ).

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The only extension of the Buser-Sarnak estimate we are aware of is due to B. Muetzel [Mue10],who recently proved a similar result with η(γ) = 2. Contrary to Theorem 1.5, we do not knowwhether the result of Corollary 1.2 is sharp regarding the number of independent lattice vectors ofnorm at most ∼

√log γ.

To prove Theorem 1.4 we consider naturally embedded graphs which capture a part of the topol-ogy and geometry of the surface, and recursively use on these graphs a systolic inequality due toB. Bollobas, E. Szemeredi and A. Thomason [BolTho97, BolSze02]. Then, we derive Theorem 1.5in the absence of a lower bound on the homological systole. In the hyperbolic case (restrictingourselves to hyperbolic metrics in our constructions), we further obtain a crucial property for theproof of Corollary 1.2: the loops given by Theorem 1.5 have embedded collars of uniform width.To prove Theorem 1.6, we adapt known constructions of surfaces with large homological systole toobtain closed hyperbolic surfaces of large genus which asymptotically approach the limit case.

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2. Local systolic geometry

In this second part of our report we present local systolic properties of some special metrics.All these metrics share the same property: they have a lot of closed geodesics whose length isprecisely their systole. These results are included in the articles [1], [5], [7] and [8], the article [1]being written in collaboration with J.C. Alvarez Paiva while the article [7] was a collaboration withC. Croke and M. Katz. In a first part we consider the two-sphere endowed with two remarkablemetrics: a singular metric with three conical singularities which was conjectured by E. Calabi torealize the minimum value of the ratio area/sys2 over smooth metrics, and the round metric. Thenwe enlarge our study using contact geometry and give strong local systolic properties for any ZollFinsler smooth metric on a closed manifold. In the sequel, otherwise stated, all Riemannian orFinsler metrics are supposed to be smooth.

2.1. Local systolic and diastolic geometry of the two-sphere. Given a smooth Riemannianmetric g on the two-sphere, recall that the systole is defined as the shortest length of a nontrivialclosed geodesic and is denoted by sys(S2, g). According to Croke’s inequality (1.1) the area isbounded from below by the systole as follows:

area(S2, g) ≥ 1/312 · sys(S2, g)2.

The constant in the inequality was successively improved by A. Nabutovsky and R. Rotman[NabRot02], S. Sabourau [Sab04] and R. Rotman [Rot06]. The best known constant (due to R. Rot-man) is 1/32 but the optimal constant remains unknown. Surprisingly, the round sphere is nota global minimum of the systolic area—the ratio area/sys2—. Indeed E. Calabi, see [Cro88], hasremarked that the singular metric gc on S2 defined by taking two identical equilateral trianglesglued along their boundary has systolic area 1/(2

√3). While gc is nor smooth or (strictly) convex,

the surface (S2, gc) can be thickened to yield convex smooth two-spheres for which the systolic areais strictly smaller than 1/π, the value which corresponds to the systolic area of the round metricg0. This singular metric gc is conjectured to achieve the global minimum of the systolic area forS2:

Conjecture 2.1 (Optimal systolic inequality for the two-sphere). For any smooth Riemannianmetric g on the two-sphere,

area(S2, g) ≥ 12√

3· sys(S2, g)2.

In section 1.1, we presented a diastolic inequality obtained in [6] with S. Sabourau that assertsthat for any smooth Riemannian two-sphere (S2, g)

area(S2, g) ≥ C · dias(S2, g)2

where C is some positive constant. Recall that the diastole denoted by dias(S2, g) is defined as thevalue obtained by a minimax process over the space of one-cycles and satisfies

dias(S2, g) ≥ sys(S2, g).

Furthermore the two quantities coincide for smooth strictly convex Riemannian metrics by [CalCao92].The diastolic area—the ratio area/dias2—of the singular metric gc equals to 1/(2

√3) and the fol-

lowing conjecture appears rather natural.

Conjecture 2.2 (Optimal diastolic inequality for the two-sphere). For any Riemannian two-sphere(S2, g),

area(S2, g) ≥ 12√

3· dias(S2, g)2.

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Observe that this conjecture is stronger than conjecture 2.1. By Pal’s Theorem [Pal21], conjec-ture 2.2 holds for the special set of singular metrics obtained by gluing two copies of any convexdisk of the plane along their boundary.

In this section we focus on local properties of systolic and diastolic area near the singular metricgc and near the round metric g0. By the uniformization theorem the two-sphere admits only oneconformal structure up to diffeomorphism. So we reduce our study to the local behaviour of systolicand diastolic area over the space of metrics conformal respectively to gc or g0. Furthermore, thesefunctionals being scale invariant, we use this additional symmetry to further reduce the space ofmetrics studied in the neighbourhood of g0, see below. We already point out that we will obtainedanother type of local result for the round metric—and more generally for Zoll Finsler manifolds—by using contact geometry and perturbation theory in section 2.3.

We begin by presenting our results in the neighbourhood of the singular metric. Denote by Mc

the space of Riemannian metrics of class C1 with three conical singularities of angle 2π/3. Weremark in [5, Proposition 2.2] that the infimum of the diastolic area over Mc coincides with theinfimum of the diastolic area over the space of smooth metrics. So Mc is relevant for our localstudy. The natural topology on Mc is the C1-topology, defined as the topology induced by theC1 compact-open topology on C1(S2,R∗+) (see [Hir76, p. 34]). In [5] we prove the following localversion of conjecture 2.2.

Theorem 2.1 (Local minimality of the singular metric for the C1-topology). There exists an openneighborhood O of gc in Mc with respect to the C1-topology such that for all g ∈ O,

area(S2, g) ≥ 12√

3dias(S2, g)2

with equality if and only if g is isometric to some multiple of gc.

As dias ≥ sys, Theorem 2.1 remains valid with the systole replacing the diastole. Our proof isbased on the study of a degree 3 ramified cover of S2 by the two-torus T2 and an optimal systolicinequality on the torus due to C. Loewner. In [Sab10], S. Sabourau found an alternative proof ofour theorem which does not make use of the uniformization theorem, but is based on the studyof the same ramified cover, and carries over to metrics which are closed enough for the Lipschitzdistance (in particular thus authorizing small variations of the angles of the three conical points).

We now present our results in the neighbourhood of the round metric. Denote byM0 the spaceof smooth Riemannian metrics of the two-sphere. The space D of smooth diffeomorphism naturallyacts on M0 and both diastolic and systolic area are invariant under this action, as well as underthe action of R∗+ on M0 by homothety : for any metric g ∈ M0, any diffeomorphism φ ∈ D andany scalar λ > 0, the metric λ · φ∗g has the same diastolic (respectively systolic) area as g. Denoteby

T0 :=f2 · g0 | f ∈ C∞(S2,R∗+) with

∫S2

fdvg0 = 4π

the space of normalized Riemannian metrics smooth conformal to g0. (Here dvg0 denotes theRiemannian volume associated to g0 which satisfies

∫S2 dvg0 = 4π.) The space T0 parametrizes the

quotient ofM0 under the action of D and R∗+, any smooth metric being isometric to some multipleof a metric in T0 thanks to the uniformization theorem. It is important to remind that diastole andsystole coincide for smooth strictly convex metrics according to [CalCao92]. Thus near the roundmetric it is equivalent to study systolic or diastolic area. In order to state our next result, we needthe following definition.

17

Definition 2.1 (Natural paths between metrics). Let g and h be two Riemannian metrics on somemanifold. The natural path gtt∈[0,1] of Riemannian metrics from g to h is defined by the followingequation on the associated distance functions :

dgt = (1− t)dg + tdh

for t ∈ [0, 1].

A natural path between two metrics is thus the linearly parametrized segment between thecorresponding distance functions. By a result of V. Guillemin [Gui76], there exist plenty of smoothdeformations of the round metric by smooth (non-isometric) Zoll metrics—metrics all of whosegeodesics are closed and of the same length—. As Zoll metrics all have the same diastolic area,the round metric can not be a strict local minimum of diastolic area. Somewhat surprising, aconsequence of our results in [8] is that the round metric is a strict local minimum of diastolic areaalong natural paths in T0. More precisely, we have the following theorem.

Theorem 2.2 (Local minimality of the round metric along natural paths). Let g be a metric inT0 different from g0 and denote by gt the natural path from g0 to g. Then there exists a positiveε such that for 0 < t < ε

area(S2, gt) >1π

dias(S2, gt)2.

This statement strongly suggests that the round metric is a local minimum of diastolic (and thussystolic) area. However recall that even in the finite dimensional case we can construct functionswith a local minimum along natural paths which is not a local minimum.

We now briefly explain how to derive Theorem 2.2 from the main result of [8].

Proof. Recall that (see [8, Proposition 4]) if Ψ : S2 → R is a non-trivial smooth function such that∫S2

Ψdvg0 = 0,

then there exists a positive ε such that for 0 < |t| < ε

area(S2, (1 + tΨ)2 · g0) >1π

dias(S2, (1 + tΨ)2 · g0)2.

Now for any g = f2 · g0 ∈ T0 the natural path from g0 to g writes as

gt = (1 + t(f − 1))2 · g0

with∫S2(f − 1)dvg0 = 0, and Theorem 2.2 directly follows.

In [8] we derive the criticality of the metric g0 for diastolic area. But we will present this resultin a more general framework in section 2.3. In particular, we will see that in fact any smooth ZollFinsler metric is a critical point of systolic area.

Both Theorems 2.1 and 2.2 are proved using the same strategy. For the variations of the metricin concern, the area could only increase while the average of the lengths of closed geodesics realizingthe diastole is constant. So at least one of these lengths is not greater than the initial diastole andthe local study of this closed curve permits to construct a short family of one-cycles sweeping outthe deformed two-sphere.

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2.2. Systole and diameter of Zoll metrics on the two-sphere. In this section we are inter-ested in comparing the systole of a Riemannian two-sphere with another Riemannian invariant,namely the diameter. C. Croke proved in [Cro88] that any Riemannian two-sphere (S2, g) satisfiesthe inequality

sys(S2, g) ≤ 9 · diam(S2, g).

The constant in the inequality was successively improved by M. Maeda [Mae94], A. Nabutovsky andR. Rotman [NabRot02], and S. Sabourau [Sab04]. The best known constant is 4, see [NabRot02] and[Sab04], and it has been strongly believed that the optimal constant was 2 meaning that the roundmetric is optimal for the relationship between these two invariants, see [NabRot02, Introduction].

In [7] we construct in collaboration with C. Croke and M. Katz counterexamples to this conjec-tured inequality. More precisely, we prove the following.

Theorem 2.3. There exist smooth variations gt of the round metric g0 by smooth Zoll metricssuch that

sys(S2, gt) > 2 · diam(S2, gt)

for sufficiently small t > 0.

So the round metric is not optimal for the ratio sys/diam. Such families of Zoll metrics with sys >2 · diam are obtained via the theorem of V. Guillemin’s [Gui76] asserting that for any smooth oddfunction on the two-sphere there exists a smooth conformal deformation of the round metric byZoll metrics whose first derivative is precisely the prescribed odd function. By carefully choosingthe odd function, we are able to ensure that the diameter strictly decreases while the systoleremains unchanged. It is important to remark that, as Guillemin’s theorem is obtained by applyingan implicit function theorem, we have no idea of how far from 2 is the optimal constant in thesystole/diameter inequality.

2.3. Local systolic geometry of Zoll Finsler manifolds. In this section we present the resultsobtained in collaboration with J.C. Alvarez Paiva in [1]. We first introduce contact geometry as anatural setting for the study of systolic inequalities and then explain how we deduce new insightsin systolic geometry by applying basic tools in perturbation theory.

A contact manifold is a pair (X,α) consisting of a (2n+1)-dimensional manifold together with asmooth 1-form α such that the top order form α∧dαn never vanishes. In the sequel we suppose ourcontact manifolds are closed and oriented in such a way that α∧dαn > 0. The kernel of α defines afield of hyperplanes in the tangent space of X (a vector sub-bundle of TX of co-dimension one) thatis maximally non-integrable called the contact structure associated to the contact form α. Notethat if (X,α) is a contact manifold and ρ : X → R is a smooth function that never vanishes, theform ρα is also a contact form which defines the same contact structure as α. Contact manifoldscome with a natural volume:

vol(X,α) :=∫Xα ∧ dαn .

They also carry a natural vector field, the Reeb vector field Rα, defined by the equations dα(Rα, ·) =0 and α(Rα) = 1. The flow of the vector field Rα (remember our contact manifolds are all closed)is called the Reeb flow and its orbits are the Reeb orbits.

Definition 2.2. The systole of a contact manifold (X,α), which we denote by sys(X,α), is thesmallest period of any of its periodic Reeb orbits. We define the systolic volume of a contactmanifold (X,α) of dimension 2n+ 1 as the ratio

S(X,α) =vol(X,α)

sys(X,α)n+1.

19

Remark that in this definition we are implicitly assuming the existence of periodic Reeb orbitson closed contact manifolds. We can bypass this thorny issue by setting S(X,α) = 0 if there areno periodic Reeb orbits, but Weinstein’s conjecture is that they always exist. More importantly,their existence has been proved for all the contact manifolds that appear in our results, see [1].

From the viewpoint of perturbation theory, one advantage of working in the contact setting isthat we may assume that every smooth deformation of a contact manifold (X,α0) is of the formρsα0, where ρs is a smooth function on M depending smoothly on the parameter. More precisely,Gray’s stability theorem (see Theorem 2.2.2 in [Gei08]) states that given a smooth deformation αs(s ranging over some compact interval), there exists an isotopy Φs such that Φ∗sαs = ρsα0. In otherwords, we may assume that the contact structure stays fixed along the deformation. Our mainresults about local systolic geometry of contact manifolds concern the following generalization ofZoll manifolds.

Definition 2.3. A contact manifold (X,α) is said to be regular if its Reeb flow is periodic and allthe Reeb orbits have the same prime period sys(X,α).

Our first main result in [1] states that the critical points of the systolic volume are precisely theregular contact manifolds.

Theorem 2.4. A contact manifold (X,α) is regular if and only if for every smooth isosystolicdeformation the derivative of the function s 7→ vol(X,αs) vanishes at s = 0.

In studying systolic volume by perturbation techniques, we face the problem that it is not adifferentiable function. In Theorem 2.4 we bypassed this difficulty by considering smooth isosystolicdeformations—smooth deformations along which the systole remains constant—. However, one ofthe key features of the work [1] is that we are able to work with arbitrary smooth deformations,which we normalize to be volume-preserving. In order to state the next result we need the following.

Definition 2.4. A smooth deformation αs of a contact form α0 is said to be trivial if there exista smooth real-valued function λ(s) and an isotopy Φs such that αs = λ(s)Φ∗sα0.

A smooth deformation αs is said to be formally trivial if for every n ∈ N there exists a trivialdeformation α

(n)s that has n-order contact with αs at s = 0.

Observe that systolic volume is constant along trivial deformations. Our second main result in[1] is the following description of the local behaviour of systolic volume near regular manifolds.

Theorem 2.5. Let (X,αs) be a smooth deformation of a regular contact manifold (X,α0). If thedeformation is not formally trivial, then the function s 7→ S(X,αs) attains a strict local minimumat s = 0. If, on the other hand, the deformation is formally trivial, then

S(X,αs) = S(X,α0) +O(|s|k) for all k > 0.

The main ingredient in proving Theorems 2.4 and 2.5 is the following. [1, Theorem 4.1] impliesthat if a deformation of a regular contact form α0 is of the form αs = Φ∗sρsα0, where Φs is anisotopy and ρs is a smooth one-parameter family of smooth integrals of motion—functions thatare invariant under the Reeb flow of α0—, then the systolic volume of αs attains a minimum ats = 0. Given the large number of isotopies and integrals of motion for a periodic Reeb flow, wecould hope that every deformation of α0 is of this form and thus prove the local systolic minimalityof regular contact manifolds. Of course, this idea does not work. However, the theory of normalforms—which we adapt to contact geometry—tells us that it almost works.

We shall now explain the consequences for classical systolic geometry. It is well known thatgeodesic flows of Riemannian and Finsler metrics are Reeb flows (see, for example, Theorem 1.5.2in [Gei08]). The precise setup is as follows: through the Legendre transform, a (not necessarilyreversible) Finsler metric F on a manifold M gives rise to a Hamiltonian H defined in the cotangent

20

bundle. The restriction α of the canonical one-form to the unit cotangent bundle S∗HM (i.e., theset of covectors where H = 1) is a contact form and its Reeb flow is the geodesic flow of the metric.A periodic Reeb orbit in (S∗HM,α) projects down to a closed geodesic on M whose length equalsthe period (and the action) of the orbit. In particular, sys(S∗HM,α) = sys(M,F ). If the metricF is Riemannian and the manifold has dimension m, the Riemannian volume of (M,F ) and thecontact volume of (S∗HM,α) are related by the equality

vol(S∗HM,α) = m!bmvol(M,F ) ,

where bm is the volume of the m-dimensional Euclidean unit ball. When the metric is Finsler,the Holmes-Thompson volume is defined by the preceding equality (see [Tho96] and [AlvTho04]for a detailed discussion of this definition). Once we remark that the Finsler metric F is Zoll—allgeodesics are closed and of the same length—if and only if the restriction of the canonical one-form to the cotangent bundle S∗HM is regular, we deduce from Theorems 2.4 and 2.5 the followingresults.

Theorem 2.6. A closed Finsler manifold (M,F0) is Zoll if and only if for every smooth isosystolicdeformation Fs the derivative of the function s 7→ vol(M,Fs) vanishes at s = 0.

Theorem 2.7. Let (M,Fs) be a smooth volume-preserving Finsler deformation of a Zoll manifold(M,F0). If the deformation is formally trivial—that is, if for every n ∈ N there exists a deformationF

(n)s by Zoll Finsler metrics that has n-order contact with Fs at s = 0—, then

sys(M,Fs) = sys(M,F0) +O(|s|k) for all k > 0.

If, on the other hand, the deformation is not formally trivial, then the function s 7→ sys(M,Fs)attains a strict local maximum at s = 0.

Our methods work better for Finsler (eventually non-reversible) metrics than for Riemannianmetrics because only the former are stable under small contact perturbations.

One of the major open problems in systolic geometry is to determine whether the canonical Rie-mannian metric in RPm (m > 2) is a minimum of the systolic volume. Specializing Theorem 2.7 tothe Riemannian setting and using the solution of the infinitesimal Blaschke conjecture by R. Michel[Mich73] and C. Tsukamoto [Tsu81], we obtain in [1] the following result.

Theorem 2.8. Let gs be a smooth volume-preserving deformation of the canonical metric on oneof the projective spaces RPm, CPm, HPm or CaP 2 where m ≥ 2. If at s = 0 the deformation gsis not tangent to all orders to trivial deformations (i.e., to deformations of the form φ∗sg0 for someisotopy φs), then the infimum of the lengths of periodic geodesics of the metric gs attains π as astrict local maximum at s = 0.

Spheres admit non-trivial Zoll deformations so the situation is more delicate. However, on thetwo-sphere the result takes a particularly simple form.

Theorem 2.9. Let g0 be the canonical metric on the two-sphere and let gs = eρsg0 be any smoothvolume preserving deformation. If dρs/ds |s=0 is not odd, then the length of the shortest periodicgeodesic of (S2, gs) attains 2π as a strict local maximum at s = 0.

This result is sharp: the main theorem of [Gui76] states that if ρ is odd, then there exists asmooth deformation eρsg0 by Zoll metrics satisfying dρs/ds |s=0 = ρ. The length of the shortestperiodic geodesic is then constantly equal to 2π along the deformation.

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3. Systolic geometry of homology classes and groups

Using systolic geometry one can construct invariants of homology classes and groups. In thislast part of our report we present some properties about these invariants. The results presentedhere are part of the article [2] written in collaboration with I. Babenko and the article [4] writtenin collaboration with H. Parlier and S. Sabourau.

3.1. Systolic volume of homology classes. Let G be a finitely presentable group, and a ∈Hm(G,Z) a non-trivial homology class of dimension m ≥ 1. We consider the various ways thisclass can be realized by a pseudomanifold endowed with a polyhedral metric. For such realizations,the two main geometrical ingredients are the volume of the pseudomanifold and the length of loopsrepresenting non-trivial elements of G. The systolic volume turns out to be the simplest naturalway to compare these geometrical quantities in order to form an invariant and is defined as follows.A geometric cycle (X, f) representing a is a pair (X, f) consisting of an orientable pseudomanifoldX of dimension m and a continuous map f : X → K(G, 1) such that f∗[X] = a where [X] denotesthe fundamental class of X and K(G, 1) the Eilenberg-MacLane space. The representation is saidto be normal if in addition the induced map f] : π1(X)→ G is an epimorphism. Given a geometriccycle (X, f), we can consider for any polyhedral metric g on X (see [Bab06]) the relative homotopicsystole denoted by sysf (X, g) and defined as the least length of a loop γ of X whose image underf is not contractible. The systolic constant of the geometric cycle (X, f) is then the value

Sf (X) := infg

vol(X, g)sysf (X, g)m

,

where the infimum is taken over all polyhedral metrics g on X and vol(X, g) denotes the m-dimensional volume of X. In the case where f : X → K(π1(X), 1) is the classifying map (inducedby an isomorphism between the fundamental groups), we simply denote by S(X) the systolicconstant of the pair (X, f). From [Gro83, Section 6], we have for any m ≥ 1 that

σm := inf(X,f)

Sf (X) > 0,

the infimum being taken over all geometric cycles (X, f) representing a non trivial homology classof dimension m. The following notion was introduced by M. Gromov in [Gro83, Section 6]:

Definition 3.1. The systolic volume of the pair (G, a) is defined as the number

S(G, a) := inf(X,f)

Sf (X),

where the infimum is taken over all geometric cycles (X, f) representing the class a.

Any integer class is representable by a geometric cycle, see Theorem 3.1 below. The systolicvolume of (G, a) is thus well defined and satisfies S(G, a) ≥ σm. But it is not clear if the infimumvalue S(G, a) is actually a minimum and what is the structure of a geometric cycle that mightachieve it. In the case where the homology class a is representable by a manifold, we know thatthe systolic volume coincides with the systolic constant of any normal representation of a by amanifold, see [Bab06, Bab08, Bru08]. A manifold is an example of admissible pseudomanifold,that is a special type of pseudomanifolds for which any element of the fundamental group can berepresented by a curve not going through the singular locus of X. In the article [2] written incollaboration with I. Babenko, we first prove the following result.

Theorem 3.1. Let G be a finitely presentable group and a ∈ Hm(G,Z) a homology class of dimen-sion m ≥ 3. For any normal representation of a by an admissible geometric cycle (X, f),

S(G, a) = Sf (X).

Furthermore, there always exists a normal representation of a by an admissible geometric cycle.

24

Here an admissible geometric cycle (X, f) stands for a geometric cycle whose pseudomanifoldX is admissible. Thus the infimum in the definition of systolic volume of a homology class is aminimum and the systolic constant of an admissible orientable pseudomanifold X depends only onthe image of its fundamental class f∗[X] ∈ Hm(π1(X),Z) as in the case of orientable manifolds.We will see in the sequel an example showing that the condition of normalization (that is, f] is anepimorphism between fundamental groups) can not be relaxed in our theorem. Loosely speaking,this theorem is proved as follows. Any geometric cycle representing the class a ∈ Hm(G,Z) can beused as starting point for the construction of the Eilenberg-MacLane space K(G, 1). This allows tocompare its systolic volume with the systolic volume of a fixed admissible geometric cycle normallyrepresenting the homology class. The comparison is done using techniques initiated by I. Babenkoin [Bab06]. The existence of such normal representations by admissible geometric cycles is deducedfrom a construction of singular manifolds representing homology classes due to N. Baas in [Baas73].

In order to understand the systolic volume invariant, one can ask for its distribution along thereal line. In [2] we show two new phenomena. First, the systolic volume function does not avoidarbitrarily large intervals.

Proposition 3.1. Let m ≥ 3. For any interval I ⊂ R+ of length at least σm, there exists a pair(G, a) consisting of a finitely presentable group and a homology class of dimension m such thatS(G, a) ∈ I.

Secondly, there is no finiteness result in great generality for systolic volume in dimension m ≥ 3.In order to give this statement content, we introduce the following definition. A class a ∈ Hm(G,Z)is said reducible if there exists a proper subgroup H ⊂ G and a class b ∈ Hm(H,Z) such thati∗(b) = a where i denotes the canonical inclusion. Otherwise the class will be said irreducible.

Theorem 3.2. For any dimension m ≥ 3 there exists an infinite sequence of finitely presentablepairwise distinct groups Gi for which at least one irreducible class ai ∈ Hm(Gi,Z) satisfiesS(Gi, ai) ≤ 1

This theorem is proved by explicitely constructing the pairs (Gi, ai) as reductions modulo primenumbers of the pair (Zm, [Tm]). This implies the following unexpected result using surgery onrepresentations by manifolds for dimensions at least 4.

Corollary 3.1. For any dimension m ≥ 4, there exists an infinite number of irreducible orientablemanifolds M of dimension m with pairewise non-isomorphic fundamental groups such that S(M) ≤1.

So we have to introduce topological or algebraic restrictions in order to derive finiteness results.For instance, given a finitely presentable group G, a homology class a ∈ Hm(G,Z) and a positivenumber T , the number of integer multiple classes ka whose systolic volume is less than T is at mostT. lnT (up to some multiplicative constant). More precisely we show in [2, section 5] the following.

Theorem 3.3. Let G be a finitely presentable group and a ∈ Hm(G,Z) where m ≥ 3. There existsa positive number C(G, a) depending only on the pair (G, a) such that

S(G, ka) ≤ C(G, a) · k

ln(1 + k)

for any integer k ≥ 1. In particular,

limk→∞

S(G, ka)k

= 0.

The proof of this result relies on the behaviour of systolic volume of geometric cycles underthe operation of connected sum, and is related to a previous collaboration with I. Babenko, see

25

[BabBal05]. It shows that systolic volume of multiples of a class is a sublinear function. For classesa whose simplicial volume is not zero, we know after Gromov [Gro83] that there exists a positivenumber C ′(G, a) depending only on the pair (G, a) such that

(3.1) S(G, ka) ≥ C ′(G, a) · k

(ln(1 + k))m.

Moreover, for fundamental groups πl of orientable surfaces Σl of genus l ≥ 1 and their correspondingfundamental classes [Σl], we know by [Gro83] and [BusSar94] that

S(πl, k[Σl]) ∼k

(ln(1 + k))2.

where f ∼ g means that there exists some positive constants c and C such that c.f ≤ g ≤ C.f .This naturally leads to the following:

Conjecture 3.1. Let G be a finitely presentable group and a ∈ Hm(G,Z) a class of non-zerosimplicial volume where m ≥ 3. Then

S(G, ka) ∼ k

(ln(1 + k))m.

The dependence of the systolic volume on torsion is another natural question. In [Gro96] Gromovmentions that it may be possible to use the torsion ofH∗(π1(M),Z) to bound from below the systolicvolume of a manifold M . Given a finitely presentable group G and a homology class a of dimensionm, we define the 1-torsion of the class a as the integer

t1(a) := min(X,f)|TorsH1(X,Z)|,

where the minimum is taken over the set of geometric cycles (X, f) representing the class a and|TorsH1(X,Z)| denotes the number of torsions elements in the first homology group of X. We nowstate the main result of [2, section 6] :

Theorem 3.4. Let G be a finitely presentable group and a ∈ Hm(G,Z) where m ≥ 2. Then

S(G, a) ≥ Cmln t1(a)

exp(C ′m√

ln(ln t1(a)),

where Cm and C ′m are two positive numbers depending on m.In particular for any ε > 0

S(G, a) ≥ (ln t1(a))1−ε

if t1(a) is large enough.

This result is optimal in the following sense: for any dimension m, there exists a sequence ofgroups Gn and homology classes an ∈ Hm(Gn,Z) such that

limn→∞

S(Gn, an)ln t1(an)

= 0.

In general the 1-torsion of a class is difficult to compute. In the case of Zn := Z/nZ, we can boundfrom below the 1-torsion of any generator by the number n. In particular, the fundamental classesof lens spaces Lm(n) realize exactly the generators of the group H2m+1(Zn,Z) and we obtain that

S(Lm(n)) ≥ (lnn)1−ε

for any ε > 0 if n is large enough.

Theorem 3.4 is proved by first bounding from below the minimal number of 2-simplices of anygeometric cycle (X, f) representing a by its 1-torsion, and then using an estimate due to M. Gromovof the systolic volume S(G, a) by the simplicial height of a—the minimal number of simplices (of

26

all dimensions) of a geometric cycle representing the class a—, see [Gro83, 6.4.C”] and [Gro96,3.C.3]. Theorem 3.4 allows us to derive the following result.

Theorem 3.5. There exists two positive constants a and b such that, for any manifold M ofdimension 3 with finite fundamental group,

S(M) ≥ a ln |π1(M)|exp(b

√ln(ln |π1(M)|)

,

where |π1(M)| denotes the cardinal of π1(M).

Now we present the particular case of the Heisenberg group of dimension 3. We obtain a newillustration of the possible behaviour of the systolic volume of cyclic coverings. The Heisenberggroup H of dimension 3 is the group of triangular matrices

1 x z0 1 y0 0 1

| x, y, z ∈ R

.

The subset H(Z) of H composed of matrices with integer coefficients (i.e. matrices for whichx, y, z ∈ Z) is a lattice, and we will denote by MH = H/H(Z) the corresponding quotient space.First of all, we obtain the following explicit upper bound for the systolic volume of multiples of thefundamental class of MH, see [2].

Theorem 3.6. Let a = [MH] ∈ H3(H(Z),Z) be the fundamental class of MH. Then

S(H(Z), ka) ≤ 19 ·S(H(Z), a)

for any integer k ≥ 1.

The constant appearing here is the one involved in the resolution of the classical Waring problem(see [BDD86]): any integer number decomposes into a sum of at most 19 fourth powers. The idea ofusing the solution of the Waring problem in order to bound from above the function S(G, ka) when(G, a) = (H(Z), [MH]) carries over to any pair (G, a) where G is a nilpotent graded group withouttorsion and a denotes the fundamental class of the corresponding nilmanifold, see [2, Theorem 7.2].

Now consider the sequence of lattices Hn(Z)∞n=1 of H, where Hn(Z) denotes the subset ofmatrices whose integer coefficients satisfy x ∈ nZ and y, z ∈ Z. Denote by MHn = H/Hn(Z) thecorresponding nilmanifolds. The manifold MHn is a cyclic covering with n sheets of MH, and thetechniques involved in the proof of Theorem 3.3 implies that

S(MHn) ≤ C · n

ln(1 + n).

The fact that the function S(MHn) goes to infinity is a consequence of Theorem 3.4.

Corollary 3.2. The function S(MHn) satisfies the following inequality:

S(MHn) ≥ a lnnexp(b

√ln(lnn)

,

where a an b are two positive constants. In particular,

limn→+∞

S(MHn) = +∞.

Note that in this case ‖MHn‖∆ = 0 and the lower bound 3.1 does not apply. For any integer nthe manifold MHn gives a non-normal realization of the class n[MH]. So normalization conditionin Theorem 3.1 cannot be relaxed.

27

3.2. Systolic area of groups. Let G be a finitely presentable group and consider the various waysthis group can be realized as the fundamental group of a finite simplicial complexe of dimension2. Using the systolic geometry of these realizations, one defines an invariant of the group calledsystolic area and introduced by M. Gromov in [Gro96, 3.C.8].

Definition 3.2. The systolic area of G is defined as

S(G) = infX

S(X),

where the infimum is taken over all finite 2-complexes X with fundamental group isomorphic to G.

If the group is free, its systolic area is zero. In converse any group which is not free has positivesystolic area by [Gro83, 6.7.A]. The best known lower bound is due to Y. Rudyak and S. Sabourauin [RudSab08] who proved that

S(G) ≥ π

16for such groups. In the same article they also prove a finiteness result for systolic area of groups.

In [4] we prove the analog of Gromov’s log γ bound (1.2) for systolic area of groups. Recall thatthe first Betti number of a group G is defined as the dimension of its first real homology group

H1(G,R) := H1(K(G, 1),R),

where K(G, 1) denotes the Eilenberg-MacLane space associated to G. Using the techniques ex-plained in section 1.3 we proved with H. Parlier and S. Sabourau in [4] the following result.

Theorem 3.7. Let G be a finitely presentable nontrivial group with no free factor isomorphic to Z.Then

S(G) ≥ C b1(G) + 1(log(b1(G) + 2))2

for some positive universal constant C.

Consider the free product Gn = Fn ∗G, where Fn is the free group with n generators and G is afinitely presentable nontrivial group. The first Betti number of Gn goes to infinity with n, while itssystolic area remains bounded by the systolic area of G. This example shows that a restriction onthe free factors is needed in the previous theorem. The order of the bound in the previous theoremis asymptotically optimal, see [4, example 7.4].

We can also bound from below the systolic area of such groups by the number of torsion elementsin their first homology group. Indeed, we prove with I. Babenko in [2] the following result.

Theorem 3.8. Let G be a finitely presentable group with no free factor isomorphic to Z. Then

S(G) ≥ C ln |TorsH1(G,Z)|exp(C ′

√ln(ln |TorsH1(G,Z)|)

,

where C and C ′ are two positive numbers.In particular for any ε > 0

S(G) ≥ (ln |TorsH1(G,Z)|)1−ε

if |TorsH1(G,Z)| is large enough.

This result should be compared with Theorem 3.4.

29

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